Solve 3 X 2 + 17 X − 6 = 0 3x^2 + 17x - 6 = 0 3 X 2 + 17 X − 6 = 0 .Given The Work Shown:${ \begin{array}{r} 3x^2 + 18x - X - 6 = 0 \ 3x(x + 6) - 1(x + 6) = 0 \ (3x - 1)(x + 6) = 0 \ 3x - 1 = 0 \quad \text{or} \quad X + 6 = 0 \end{array} }$Based On The Work Shown, Which

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $3x^2 + 17x - 6 = 0$, using a step-by-step approach. We will also analyze the work shown and provide a clear explanation of the solution.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Given Equation

The given equation is $3x^2 + 17x - 6 = 0$. To solve this equation, we will use the method of factoring. Factoring involves expressing the quadratic expression as a product of two binomials.

Step 1: Factor the Quadratic Expression

The first step in solving the equation is to factor the quadratic expression. We can start by looking for two numbers whose product is $3 \times (-6) = -18$ and whose sum is $17$. These numbers are $18$ and $-1$, so we can rewrite the equation as:

$$3x^2 + 18x - x - 6 = 0$$

Now, we can factor the expression by grouping:

$$3x(x + 6) - 1(x + 6) = 0$$

Step 2: Factor the Expression Further

We can factor the expression further by recognizing that both terms have a common factor of $(x + 6)$. Therefore, we can write:

$$(3x - 1)(x + 6) = 0$$

Step 3: Solve for $x$

Now that we have factored the expression, we can solve for $x$ by setting each factor equal to zero:

$$3x - 1 = 0 \quad \text{or} \quad x + 6 = 0$$

Solving for $x$ in each equation, we get:

$$x = \frac{1}{3} \quad \text{or} \quad x = -6$$

Conclusion

In conclusion, we have solved the quadratic equation $3x^2 + 17x - 6 = 0$ using the method of factoring. We started by factoring the quadratic expression, then factored the expression further, and finally solved for $x$ by setting each factor equal to zero. The solutions to the equation are $x = \frac{1}{3}$ and $x = -6$.

Discussion

The work shown in the problem is a clear and concise step-by-step approach to solving the quadratic equation. The use of factoring allows us to simplify the equation and solve for $x$ in a straightforward manner. This method is particularly useful when the quadratic expression can be easily factored.

Common Mistakes

When solving quadratic equations, it's essential to be careful when factoring the expression. One common mistake is to forget to factor the expression completely, leading to incorrect solutions. Another mistake is to solve for $x$ incorrectly, resulting in incorrect values.

Real-World Applications

Quadratic equations have numerous real-world applications, including physics, engineering, and economics. For example, the motion of an object under the influence of gravity can be modeled using quadratic equations. Similarly, the cost of producing a product can be represented using quadratic equations.

Final Thoughts

Solving quadratic equations is a fundamental skill that requires practice and patience. By following the step-by-step approach outlined in this article, you can develop a deeper understanding of quadratic equations and improve your problem-solving skills. Remember to be careful when factoring the expression and solving for $x$, and don't hesitate to seek help if you're struggling.

Additional Resources

• Quadratic Formula
• Factoring Quadratic Expressions
• Solving Quadratic Equations

Glossary

• Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two.
• Factoring: Expressing a quadratic expression as a product of two binomials.
• Binomial: A polynomial expression with two terms.
• Quadratic Formula: A formula used to solve quadratic equations, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

References

• [1] "Quadratic Equations" by Math Open Reference. [Online]. Available: https://www.mathopenref.com/factoringquadratic.html.
• [2] "Solving Quadratic Equations" by Khan Academy. [Online]. Available: https://www.khanacademy.org/math/algebra/x2f1f4f/solving-quadratic-equations/v/solving-quadratic-equations.
• [3] "Quadratic Formula" by Wikipedia. [Online]. Available: https://en.wikipedia.org/wiki/Quadratic_formula.
  Quadratic Equations Q&A ==========================

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: What is the difference between factoring and the quadratic formula?

A: Factoring involves expressing the quadratic expression as a product of two binomials, while the quadratic formula involves using a formula to solve the equation. Factoring is often easier and more intuitive, but the quadratic formula can be used to solve any quadratic equation.

Q: Can I use the quadratic formula to solve a quadratic equation that can be factored?

A: Yes, you can use the quadratic formula to solve a quadratic equation that can be factored. However, it's often easier and more efficient to factor the equation directly.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Forgetting to factor the expression completely
• Solving for $x$ incorrectly
• Not checking the solutions to make sure they are valid
• Not considering the possibility of complex solutions

Q: How do I check my solutions to make sure they are valid?

A: To check your solutions, you can plug them back into the original equation and make sure they are true. You can also use the quadratic formula to check your solutions.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including physics, engineering, and economics. For example, the motion of an object under the influence of gravity can be modeled using quadratic equations. Similarly, the cost of producing a product can be represented using quadratic equations.

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some tips for solving quadratic equations?

A: Some tips for solving quadratic equations include:

• Using the quadratic formula as a last resort
• Factoring the expression completely before solving for $x$
• Checking your solutions to make sure they are valid
• Considering the possibility of complex solutions

Q: Can I use technology to solve quadratic equations?

A: Yes, you can use technology to solve quadratic equations. Many graphing calculators and computer algebra systems can be used to solve quadratic equations.

Q: What are some common types of quadratic equations?

A: Some common types of quadratic equations include:

• Quadratic equations with real solutions
• Quadratic equations with complex solutions
• Quadratic equations with rational solutions
• Quadratic equations with irrational solutions

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including physics, engineering, and economics.

Q: What are some real-world examples of quadratic equations?

A: Some real-world examples of quadratic equations include:

• The motion of an object under the influence of gravity
• The cost of producing a product
• The area of a circle
• The volume of a sphere

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some tips for teaching quadratic equations?

A: Some tips for teaching quadratic equations include:

• Using real-world examples to illustrate the concept
• Providing multiple examples and practice problems
• Encouraging students to use technology to solve quadratic equations
• Emphasizing the importance of checking solutions to make sure they are valid

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including physics, engineering, and economics.

Q: What are some common mistakes to avoid when teaching quadratic equations?

A: Some common mistakes to avoid when teaching quadratic equations include:

• Not providing enough examples and practice problems
• Not emphasizing the importance of checking solutions to make sure they are valid
• Not using real-world examples to illustrate the concept
• Not encouraging students to use technology to solve quadratic equations

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some real-world applications of quadratic equations in physics?

A: Quadratic equations have numerous real-world applications in physics, including:

• The motion of an object under the influence of gravity
• The trajectory of a projectile
• The oscillations of a spring
• The vibrations of a guitar string

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including engineering and economics.

Q: What are some common types of quadratic equations in physics?

A: Some common types of quadratic equations in physics include:

• Quadratic equations with real solutions
• Quadratic equations with complex solutions
• Quadratic equations with rational solutions
• Quadratic equations with irrational solutions

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some real-world examples of quadratic equations in engineering?

A: Some real-world examples of quadratic equations in engineering include:

• The design of a bridge
• The construction of a building
• The optimization of a system
• The analysis of a circuit

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including physics and economics.

Q: What are some common types of quadratic equations in engineering?

A: Some common types of quadratic equations in engineering include:

• Quadratic equations with real solutions
• Quadratic equations with complex solutions
• Quadratic equations with rational solutions
• Quadratic equations with irrational solutions

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some real-world applications of quadratic equations in economics?

A: Quadratic equations have numerous real-world applications in economics, including:

• The cost of producing a product
• The demand for a product
• The supply of a product
• The price of a product

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including physics and engineering.

Q: What are some common types of quadratic equations in economics?

A: Some common types of quadratic equations in economics include:

• Quadratic equations with real solutions
• Quadratic equations with complex solutions
• Quadratic equations with rational solutions
• Quadratic equations with irrational solutions

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some real-world examples of quadratic equations in finance?

A: Some real-world examples of quadratic equations in finance include:

• The calculation of interest rates
• The determination of investment returns
• The analysis of stock prices
• The optimization of portfolios

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including physics and engineering.

Q: What are some common types of quadratic equations in finance?

A: Some common types of quadratic equations in finance include:

• Quadratic equations with real solutions
• Quadratic equations with complex solutions
• Quadratic equations with rational solutions
• Quadratic equations with irrational solutions

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, including algebra, geometry, and trigonometry.

Q: What are some real-world applications of quadratic equations in computer science?

A: Quadratic equations have numerous real-world applications in computer science, including:

• The optimization of algorithms
• The analysis of data structures
• The design of computer networks
• The development of artificial intelligence

Q: Can I use quadratic equations to solve problems in other areas of science?

A: Yes, quadratic equations can be used to solve problems in other areas of science, including physics and engineering.

Q: What are some common types of quadratic equations in computer science?

A: Some common types of quadratic equations in computer science include:

• Quadratic equations with real